Mixing layer instability and vorticity amplification in a creeping viscoelastic flow

We report quantitative evidence of mixing-layer elastic instability in a viscoelastic fluid flow between two widely spaced obstacles hindering a channel flow at $\text{Re}\ll 1$ and $\text{Wi}\gg 1$. Two mixing layers with nonuniform shear velocity profiles are formed in the region between the obstacles. The mixing-layer instability arises in the vicinity of an inflection point on the shear velocity profile with a steep variation in the elastic stress. The instability results in an intermittent appearance of small vortices in the mixing layers and an amplification of spatiotemporal averaged vorticity in the elastic turbulence regime. The latter is characterized through scaling of friction factor with Wi and both pressure and velocity spectra. Furthermore, the observations reported provide improved understanding of the stability of the mixing layer in a viscoelastic fluid at large elasticity, i.e., $\text{Wi}\gg 1$ and $\text{Re}\ll 1$ and oppose the current view of suppression of vorticity solely by polymer additives.

Figure 1

Schematic of experimental setup (not to scale).

Figure 2

Normalized friction factor $f/f_{\mathrm{lam}}$ vs Wi with the fits marked by dash lines above the first instability $f\text{Wi}\sim {\text{Wi}}^{0.5}$ and in the ET regime $f\text{Wi}\sim {\text{Wi}}^{0.2}$, which yield critical values for for respective transitions ${\text{Wi}}_{c1}\simeq 12$ and ${\text{Wi}}_{c2}\approx 45$. Gray band indicates the transition region. Top inset: Friction factor $f$ versus Wi. Laminar flow is represented by scaling $f_{\mathrm{lam}}\sim {\text{Wi}}^{-1}$. Bottom inset: Frequency power spectra of pressure fluctuations $S\left(P\right)$ in log-log coordinates for several Wi values. Dashed line indicates the power spectra decay with an exponent $\beta \approx -3.2\pm 0.2$ in the ET regime.

Figure 3

Spatially and time-averaged vorticity $\overline{\omega }$ vs Wi with the fits by dashed lines $\overline{\omega }\sim {\text{Wi}}^{0.5}$, and $\overline{\omega }\sim {\text{Wi}}^{0.2}$. The gray band indicates the transition region of small vortex generation. Top inset: Frequency power spectra of cross-stream velocity, $S\left(v\right)$, for several Wi shown in linear coordinates. Bottom inset: $S\left(v\right)$ in log-log presentation for the same Wi. Dashed line indicates the power spectra decay with an exponent $\alpha \approx -3.4\pm 0.1$ in the ET regime.

Figure 4

Velocity fields between two obstacles at low (left-panel) and high (right-panel) magnifications for three values of Wi. The fields are obtained from $\mu \mathrm{PIV}$ and averaged over $\approx 1$ s. Black arrows in left panel indicate the flow direction. Black circles in right panel highlight small vortex regions that are used for circulation calculation.

Figure 5

Radial dependence of circulation $\mathrm{\Gamma }$ for three values of Wi. Inset: variation of $\left|\mathrm{\Gamma }\right|$ with Wi. The dashed line is a fit $\left|\mathrm{\Gamma }\right|\sim {(\text{Wi}-55)}^{0.2\pm 0.05}$ to the data.

Figure 6

Streamwise velocity profile $u\left(y\right)$ for (a) Newtonian solvent and [(b)–(d)] polymer solution at three Wi values. Red symbols in panels (c) and (d) indicate the inflection points on the profiles. Insets in panels (c) and (d) show the positions of inflection points in details; dashed line is a tangent at the inflection point. Noticeably, the velocity profile just above the first instability at $\text{Wi}=15.9$ is similar to that in the laminar region of a Newtonian solvent.

Figure 7

Illustration of full streamwise velocity profile $u(y/R)$ in the inner and outer regions of the obstacles to demonstrate a similarity with two mixing layers. Red symbols are inflection points on the profile and solid lines are tangents at the inflection points.